If given a rectangular sheet of cardboard and a small square of the same
size cut from each corner, while each side folds up to form a lidless box,
explore what sizes of squares can be cut from each corner to produce a box
with a certain volume. Explore what sizes of the square produce the maximum
I have chosen the length of the box to be 10 cm. and the width of the
box to be 5 cm. for the purpose of simple illustration. Let the height of
the box be x cm. As the length of x varies, the box will also vary. Click
here to view this GSP illustration.
The volume is seen along the y axis, while the length of x is seen along
the x axis. Notice that the maximum of the graph is at appromately x=3 and
that the volume is 400 at approximate values of x=1.55 and x=4.8.
As seen in the graph constructed above, we also see that the volume is
400 cubic inches when x is two different values: approximately 1.55 and
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